You did not appreciate my point. You said that "we all" agree that infinity is boundless. But as someone who's studied university math, I've seen so many examples of bounded, infinite sets that the idea that "infinity is bounded" is obviously false.

Therefor it is false that "we all agree." That was my point. So even if you disagree with me regarding infinity, that very disagreement makes my point! Because my point is that we do NOT "all agree."

That said, reading through your post it seems that you are arguing from a finitist or ultra-finitist perspective. Given that, it's perfectly sensible for you to deny infinite sets. In so doing you must also abandon a rigorous construction of the real numbers, which means you're going to lose the foundation of modern physical science along with most of modern math.

Now it's perfectly consistent to do this, and I have no objections to your making that choice. But it seems to me somewhat nihilistic, since it forces you to reject the whole of modern science along with physics, which turns out to be founded on infinitary and nonconstructive math.

So as I say I have no problem with the logical consistency of your point of view, though it does seem to limit the conversation. If you say, "Let's talk about infinity" when in fact you reject infinity, further dialog seems pointless.

All that said, you did mention a number of interesting and widely-believed fallacies and misunderstandings, which I will endeavor to correct for anyone who may be interested.

Serendipper wrote:Below, you're proceeding to talk about what you haven't defined. What do you mean when you say there are "infinite" numbers between 1 and 0 if infinity is not boundless?

As I say, if you deny that there are infinitely many real numbers between 0 and 1, that is your choice. I assume you must deny there are infinitely many rational numbers between 0 and 1 as well. This is your right, to adopt a finitist or ultra-finitist stance. It just makes conversation pointless.

When you were in high school and they showed you the real number line in Algebra I when you were 14 years old or so, did you complain that there can only be finitely many points between 0 and 1? I confess I don't understand this point of view.

And of course it is manifestly obvious that the unit interval is bounded. No member is ever less than 0 nor greater than 1. The unit interval is an infinite, bounded set.

But really, if you want to use your time on God's earth to post to an Internet forum that you think there are only finitely many points on the real number line between 0 and 1, more power to you. I won't stand in your way.

Serendipper wrote: The first step in an argument is to define terms clearly before using the terms to construct an argument, otherwise you're saying a+b+c=z but not saying what the variables are. Defining terms commits you to what you're saying.

Mathematically, a set is infinite if it may be bijected to a proper subset of itself. That was one of your dictionary definitions if I recall. Galileo noted this in the 1600's and various non-Western mathematicians noted it in the 1200's or earlier.

Serendipper wrote:Probably why 1+2+3+4+5+.... forever = -1/12

Oh my. You are a victim of some very unfortunate misinformation floating about the Internet.

The infinite series 1 + 2 + 3 + 4 + ... of course diverges. Or we can say that it converges to +infinity in the extended real numbers. Or we can say that it converges to Aleph-null as a cardinal, or omega as an ordinal. These are all different ways of saying that this infinite sum "converges to a point at infinity."

So, what is this -1/12 business about? Briefly, everyone knows that the sum of the positive integers is infinity, or undefined, whichever formalism you're using at the moment. But there's a thing called "Zeta function regularization" that says that something called zeta(-1) is -1/12. And you can choose to INFORMALLY interpret zeta(-1) as the series 1 + 2 + 3 + ... But it is NOT the same thing as that series. I'll give you some links that explain all this.

Why some ignorant Youtubers decided to start an online campaign to confuse people I have no idea. But that's modern life. The Internet, which was supposed to make us all smart, has in fact made us all much more stupid. And this -1/12 business is a classic example.

Serendipper wrote:infinity doesn't exist. If it did, the sum of all natural numbers would certainly equal it.

Oh my. As I said, it's perfectly ok for you to reject completed infinity That's finitism. You can even reject potential infinity. That's ultra-finitism. Those positions are logically consistent (though there's no known axiomatic basis for ultra-finitism). But they're nihilistic, in the sense that you have just kicked the rug out from not only math, but physics as well. I'd say that's an intellectual challenge for your position, to rebuild physics on finitistic or even constructive principles. There actually are people doing the latter.

But now it seems you are drawing your finitistic conclusion based on a misunderstanding of bad Youtube videos. Every mathematician knows that 1 + 2 + 3 + ... diverges to infinity. The interpretation of that sum as -1/12 is a distortion and abuse of a very sophisticated bit of mathematics in higher complex number theory. Somebody made a misleading video and a horde of people just ran with the wrong idea. It's awful frankly. But if that's your evidence for rejecting infinity, your argument is refuted. Please read the links I gave you above to put the -1/12 business into its proper context.

More nonsense on the same lines I imagine. The explanation of -1/12 in terms of analytic continuation and zeta function regularization is all over the Internet. For you to choose the completely wrong interpretation and then expound on your thesis of infinity based on it is truly ... well, it's your right. But I question your motivation. After all, the correct information is just as easy to find as the falsehoods.

Serendipper wrote:You're making divisions where there are none, or did you forget that you cut it? You're creating differentiations on the fly and will need an infinite amount of time to finish, so you're assuming infinity in your proof of it. This was spelled out nicely in the article I posted http://theorangeduck.com/page/infinity-doesnt-exist

Here you're again arguing that the unit interval's not an infinite set. That's not a serious intellectual position.

Serendipper wrote:Keep Adding OneThe obvious argument for infinity existing is that, given some number, it is always possible to add one to get a new number. Therefore there must be an infinite number of numbers.

The problem with this argument is that it presupposes infinity exists already. More specifically it assumes that a process can be repeated an infinite number of times. If you can't repeat a process infinitely, and there isn't infinite time, it isn't possible to continue adding one forever.

If you reject taking successors, that makes you an ultra-finitist. Fine with me. You not only reject the real numbers but the counting numbers as well. Whatever. So what happens to physics in your theory? You can't do modern physics without the real numbers and the modern theory of infinite series.

Serendipper wrote:Likewise, you're just adding new numbers between 0 and 1 and you're reliant upon infinite time to complete the process, but if you stop to inspect along the way, the answer will always be finite.

The real numbers are a mathematical abstraction. They don't require time or space or energy or any physical resource to construct.

I would agree with you that PHYSICAL infinity may well not exist. I don't believe it does. But we are talking about MATHEMATICAL infinity. Mathematical infinity does not require any physical resources to construct.

Serendipper wrote:That is why 1-1+1-1+1-1... forever = 1/2 because if you stop, you get either 0 or 1.

It's tragic that people can get so many bad ideas from the Internet, yet won't take the time to learn any actual math.

Serendipper wrote:No, you've defined it to be so. You're not observing, but conjuring.

Did you have a bad math teacher when you were young? Where is this coming from?

Serendipper wrote:You've defined the universe to be a fraction 1/x where x >1, so you've just moved the goal posts from simply counting forever to counting forever and plugging into a function... it's the same problem of using infinity to prove infinity. You need an infinite amount of x to prove there are infinite amounts between 1 and 0. Why bother? Just prove there are infinite x >1.

That, I'm afraid, is incoherent.

Serendipper wrote:You can only cut something so many times before the knife becomes larger than the thing you're cutting and that fact will become self-evident once you begin and persist in cutting.

The only way to "have" infinity is to stop the process, but if you stop the process, the result will always be finite.

How did you get these ideas? I'm curious how someone goes down this path of rejecting all of modern math and physical science.

Things could be partly infinite.For example, a road could be endless in one direction, but endful in another.MEST could in fact be infinitely divisible, but finitely multipliable, if you know what I mean, or vice versa.Just because there's an infinite amount of total apple, doesn't mean apple has to be omnipresent. Now whether anything/everything actually is infinite, is another matter.I think finitude and infinitude are equally possible.But finite, unomniscient beings such as ourselves couldn't definitely prove either-or/neither-nor.But if something appears to go on, and on...and on, we may feel like concluding it's endless.Conversely, after many unsuccessful attempts to divide a particle, we may may like concluding it's indivisible.

Serendipper wrote:Below, you're proceeding to talk about what you haven't defined.

ps to my previous post. I did a quick preliminary read of your paper that you linked. I am impressed. You have a thesis. You are wrong about a lot of things, but you did mention all the relevant topics. You've done your homework.

From what I can see so far, you keep saying you're talking about mathematical infinity, but as arguments against it you immediately cite physical analogies. But just because something is physically impossible doesn't mean we can't work with it as an abstract entity. You constantly fall back on physical arguments even when you claim to be arguing against abstract mathematical infinity.

You are absolutely correct that the nub of the matter is the Axiom of Infinity. It is indeed arbitrary, in the sense that both it and its negation are perfectly consistent with the other axioms of math. There is no absolute truth of the matter; and no logical reason to prefer one to the other.

However, there is a pragmatic, practical reason. When you assume the axiom of infinity, you can construct the real numbers and do all of modern math and physics. When you deny the axiom of infinity, you get a far more paltry universe that can't be made to serve the needs of mathematical foundations.

So ultimately the reason we accept the Axiom of Infinity is that it is more useful. We do NOT claim it's "true." You are chasing a strawman. Nobody is claiming mathematical infinity is true. Only that it's useful.

Once we accept the axiom of infinity, we get modern math and the physics that's based on it. So we assume infinity. It's working great. Is it "true?" It's hard to even know whether such a question is meaningful. Are there infinite sets? I myself question whether there are ANY sets at all. A set is an abstract object in math, very much different than the naive sets we think of as "collections of objects." Mathematical sets are much stranger than that.

The axiom of infinity is no more "true" or "false" than whether the knight in chess "really moves that way." What a ridiculous question! It's a formal game. It's not true and it's not false. The rules are what they are, and if they are consistent and interesting, we accept them. Chess is fun to play, so we play. Math is fun to play, and the physicists and engineers find it useful. That's as far as the ontology goes.

I mean, forget about an infinite set. Does the empty set exist? Ponder that.

I hope you'll take some of my points to heart. You're tilting at windmills. You think somebody thinks the axiom of infinity is true. On the contrary. People who think about the question at all, understand that the reason we accept infinity in math is because it's useful.

ps -- You may be interested to read a pair of papers by Penelope Maddy, Believing the Axioms parts I and II. You can find pdfs if you Google around. She walks through each axiom of ZFC and discusses the history and philosophy of how and why it came to be adopted.

barbarianhorde wrote:How... is it possible to argue against mathematical infinity?Is there a way of proving there can be only a finite number of numbers?

I think Peanos axioms are invalid if there is a limit somewhere.

@Serendipper is correct when he states (in his paper that he linked) that the axiom of infinity, which states that there is an infinite set, is an assumption. It can't be proven. Nor can its negation. It's simply an axiom that you can assume or deny. It gives more fruitful, useful, and interesting math to accept it, so we do. It's perfectly valid to study the system you get when you assume its negation.

Also note that even in the absence of the axiom of infinity, there are infinitely many counting numbers: 0, 1, 2, 3, 4, 5, 6, ... There just isn't a completed SET of them. In set theory without infinity, the counting numbers form a proper class: a collection that's "too big" to be a set.

There is an even more radical position, which is that there aren't even arbitrarily large finite numbers. That's ultra-finitism.

So we have:

* Standard math in which we have infinite sets;

* The Peano axioms in which we have infinitely many natural numbers, but not a set of them. That's finitism.

* The denial of arbitrarily large finite numbers. That's ultra-finitism. There is no known axiomitization of ultra-finitism. It's mostly the realm of cranks BUT there are a couple of famous serious ultra-finitists.

Serendipper wrote:In that case the cycles would be infinitely reoccurring meaning we've already had this conversation an infinite number of times and are destined to have it infinitely more. Don't you remember what I said last time? Why must we keep rehashing this throughout eternity?

Well, right off the bat, there's an infinity. Second, if there is a finite number of possible moments, then it must recycle, if not, not.

How long is a moment? You have to know how long a moment is before you can calculate how many exist.

It is basically the time taken by light of specified wavelength emitted by a Cesium 133 atom to execute 9,192,631,770 vibrations.

Right after the big bang there was no cesium 133, so how long was 1 second? Time has no intrinsic meaning but is only a relation of one thing to another thing. If it helps to see a phd physicist and philosopher say the same thing, fwd to 38:00 here https://www.youtube.com/watch?v=N-NTXoYTvao

(Btw the consensus of those 4 guys is that time is not fundamental. The guy on the right however disagrees somewhat because if the laws of the universe are emergent, then they must have evolved through time and therefore time could not be a product of the fundamental laws, yet he still doesn't see time as truly fundamental.)

There could be a seemingly infinite number of seconds in the finite span between now and the big bang just like there are seemingly infinite numbers of numbers between 0 and 1 because time is related to the very thing that was created and what we call seconds are changing the closer we get to the beginning (light moves faster), but this infinity is only an illusion produced by infinite regression once again because seconds are defined by the thing we're trying to measure.

Asking what came before the big bang is akin to asking what's north of the north pole: there is no north of the north pole and there is no before the beginning.

But more to the point and all joking aside, how could cycles take place inside infinite time if time only exists inside the cycle?

Well, again, in astrophysics - me going by what I have read, and then also going by me cornering an astrophysicist at a recent gathering - it is not consensus that time started just before the Big Expansion or that there are not other cycles - the same or otherwise - have gone before or that the Big Bang did not take place within a larger universe context. IOW that Big Bang was local. I asked specfically about infinite time and volume and neither have been ruled out and there is no clear majority either.

Truth is independent of majority consensus and, apart from grass being green and the sky blue, majority opinion is more likely to be wrong than right.

Simulations show that for most study designs and settings, it is more likely for a research claim to be false than true. Moreover, for many current scientific fields, claimed research findings may often be simply accurate measures of the prevailing bias.

Atheism is the prevailing bias and physicists will not let go of infinity until theism is eliminated because they need it as a substitute for god.

So (how did you put it...) cornering an astrophysicist at a party for his/her opinion about consensus opinion + $2 will get you a cup of brown liquid.

It's not a matter of consensus opinion, but realization that time is a relation of one thing to another thing and doesn't start until "things" come into existence just like "north of" doesn't start until we move south of the north pole.

Obviously the energy that went into winding the spring is eternal, but that's absence of time instead of infinite amounts of it. Energy doesn't observe time or space, but defines it. Energy travels at the maximum speed, which is instant ( because when you travel c, no time passes, so it's instant).

Again, if we allow for there suddenly being an extremely neg-entopic state, we cannot then rule out other hypotheses based on the second law. We allow, in this one version of the Big Bang theory, for the sudden negentropic creation of everything. Once that is one of the axioms of your belief, you cannot criticize other theories for seeming to violate the second law, since you've already done it yourself. Well, it only happened once, it not a great defense.

The 2nd law seems obviously false to me because if it were a law, then everything would have dissolved to disorder long ago and a big bang couldn't have manifested. Time in terms of order seems counter-intuitive considering that life would then be going backwards in time since life is negative entropy. How can we have life getting more ordered through the same time that other things are getting more disordered and call that a standard of measurement?

What does it mean to happen over and over outside of time and space?

Not outside, but as time and space.

I'm not sure you understand what I meant. Time exists inside the cycle and we are outside the cycle being objective observers. What does it mean for the cycle to happen over and over (when viewed outside the cycle) when "over and over" is a property only inside the cycle?

Here's an analogy: a casino puts up a display showing the history of red and black at the roulette table and people think if there are a string of reds, then black must be more likely to come next, but the events are not connected and the history means nothing. So, every event is the first event because there is nothing intrinsically linking the events.

So if a universe comes and goes, there is nothing left to be a memory and each universe is the first and only universe. The concept of "over and over" only exists when memory exists, and I don't mean human memory, but the universe would somehow remember because that information would be relevant to another process and encoded into the universe fabric. Time moves in the direction of unknown to known. A bird flaps its wings over and over because each flap is encoded into the time-space continuum, but once the universe is over, those flaps never happened because the information is forever lost. That doesn't mean the energy is lost, but only the pattern.

Time is a relation of one thing to another thing. It takes me 1/24 revolution of the earth to drive 60 miles. We relate time to the decay of atoms or the speed of light, so time is a relation of one thing to another thing and there were no things before the big bang, except the one thing that caused the big bang.

You can say this, but again, there is no consensus this is the case. And since we do not know what that 'one thing' was, we do not know what it's limits are, if it is still present, if something can reboot or reenergize what is happening nad so on.

This is all simple deduction. There can only ever be one thing. If we assert more than one thing, maybe 2 things, then how does thing #1 relate to thing #2? They would have to relate to each other or they wouldn't exist to each other and we wouldn't have 2 things. So in order to have 2 things, there must be some relation between them... and if that is so, then they are one thing. So one thing is the only possible number of things.

How does a cause influence an effect? The answer can only be that they are the same process. There is not one thing affecting another thing, but only one thing. Otherwise we'd have to explain how one thing can both relate to another thing while also not relating to another thing (mutual exclusivity is required to be a thing, otherwise it's a dissection or abstraction, but mutual exclusivity necessitates nonexistence).

So, before the big bang there was one thing and after the big bang there is still only one thing. That's from the spacial perspective. Temporally, likewise, there is only one moment. There is not an infinite number of moments throughout time, but only "now" in absence of time. Otherwise we have to explain how one moment causes the next moment without influencing it, which is ridiculous.

"The distinction between the past, present and future is only a stubbornly persistent illusion." - Albert Einstein

No doubt I have confirmation bias

You mean you're actually getting data on this??????

I suppose I could have taken the opposite position and tried to disprove what I suspect instead of confirming it. At least then I'd have everyone unwittingly on my side

It is basically the time taken by light of specified wavelength emitted by a Cesium 133 atom to execute 9,192,631,770 vibrations.

Right after the big bang there was no cesium 133, so how long was 1 second?

Cesium 133 is part of one of the ways we measure time. Of course we experience time without cesium 133 and we don't know if there were thing to relate one thing to another back then. We do not know if universes emerge from a larger universe - one of the theories out there - and so this larger context would have time.

Time has no intrinsic meaning but is only a relation of one thing to another thing. If it helps to see a phd physicist and philosopher say the same thing, fwd to 38:00 here https://www.youtube.com/watch?v=N-NTXoYTvao

(Btw the consensus of those 4 guys is that time is not fundamental. The guy on the right however disagrees somewhat because if the laws of the universe are emergent, then they must have evolved through time and therefore time could not be a product of the fundamental laws, yet he still doesn't see time as truly fundamental.)

That's all peachy, but again, in the astrophysics community there is no consensus about infinite vs. finite, in time or volume.

There could be a seemingly infinite number of seconds in the finite span between now and the big bang just like there are seemingly infinite numbers of numbers between 0 and 1 because time is related to the very thing that was created and what we call seconds are changing the closer we get to the beginning (light moves faster), but this infinity is only an illusion produced by infinite regression once again because seconds are defined by the thing we're trying to measure.

Could be. I am not arguing that the universe is infinte, though I tend to think it is, but that's just my intuition. But, sure could be. I am not saying you are wrong. I am questioning your certainty or better put, I see little reason to be certain around this issue.

Asking what came before the big bang is akin to asking what's north of the north pole: there is no north of the north pole and there is no before the beginning.

Seriously, I knew this idea 20 years ago and I am so tired of it getting trotted out by other laypeople. Yes, it seemed like there was some consensus for a while about this, but there isn't now. So trotting out old explanations that scientists and science writers used to try to help people with this counterintuitive idea THEN, does very little for me, because they are not sure about this at all.

Truth is independent of majority consensus and, apart from grass being green and the sky blue, majority opinion is more likely to be wrong than right.

Right, but you are working with opinions from the scientific community and THEIR consensus. Am I supposed to take a video with four people in it as evidence when you are here dismissing the consensus of the scientific community? A consensus of four is also fallible.

And this holds for what is below. Your ideas are based on and specifically refer to ideas trickling out of the scientific community. Now you are telling me that we shouldn't look at consensus in the science community. Now, first off, I don't always do this. But second, here you are trotting out the old semi-consensus as if it is fact. And then dismissing consensus. Fine. You just threw out both authorities.

Since I make no claims to be sure, that's really easy to live with. Since you seem to think you have proven the universe is finite, good luck with having undermined your own authorities that you appeal to.

Atheism is the prevailing bias and physicists will not let go of infinity until theism is eliminated because they need it as a substitute for god.

But many physicist let go of infinity. Now you're just making stuff up.

So (how did you put it...) cornering an astrophysicist at a party for his/her opinion about consensus opinion + $2 will get you a cup of brown liquid.

Right except I also researched what percentages believed various positions and this fit with his estimates of his peers. He wasn't telling me his stand, I don't know what that is in fact.

And again, you are the one claiming to have demonstrated that infinity is impossible. I am saying that I am skeptical. Fine, now you seem to be saying you do not base your ideas on the ideas of astrophysicists and cosmologists and don't care what the experts say. Great. I see even less reason to believe your ideas which are in part based on the positions of physicists and cosmologist. IOW their ideas are parts of your arguments.

You seem to think you can demonstrate infinity is not possible. I just don't see it. It seems very speculative to me. Especially now that you dismiss scientists, which would include those whose ideas you use. I think we should remain utterly unconvinced by your arguments, only more so now.

You did not appreciate my point. You said that "we all" agree that infinity is boundless.

Umm.... no. I said, "Hopefully we can all agree that a good working definition of infinity is "boundless", "without bounds or constraints: either physical or conceptual"."

Apparently that was too much to hope for?

Anyway, if you do not agree with the definition, then simply submit a better one. If you can't define the words you're using, then how can you claim to know what you're talking about?

I've seen so many examples of bounded, infinite sets that the idea that "infinity is bounded" is obviously false.

Those sets are bounded by identity, but unbounded otherwise. Bounded by identity means there are infinite things (unbounded numbers of things) that identify as a member of a category, so if we say infinite oranges, we do not mean infinite apples. When you say there are infinite numbers between 1 and 0, you're defining a category and claiming there exists infinite items in that category. Clearly you're suggesting the number of items in the category are unbounded, so my definition stands.

That said, reading through your post it seems that you are arguing from a finitist or ultra-finitist perspective. Given that, it's perfectly sensible for you to deny infinite sets. In so doing you must also abandon a rigorous construction of the real numbers, which means you're going to lose the foundation of modern physical science along with most of modern math.

No, I can switch between constructs just like I can switch between playing checkers and chess. I'm not confined to playing checkers with the rules of chess and likewise I'm not confined to finitism in math.

Now it's perfectly consistent to do this, and I have no objections to your making that choice. But it seems to me somewhat nihilistic, since it forces you to reject the whole of modern science along with physics, which turns out to be founded on infinitary and nonconstructive math.

That's not true. Physics says there is a smallest length called the planck length which you are violating with your 1 and 0 example. So you're the one who must abandon physics and reality to entertain this fantasy world of infinity.

So as I say I have no problem with the logical consistency of your point of view, though it does seem to limit the conversation. If you say, "Let's talk about infinity" when in fact you reject infinity, further dialog seems pointless.

I submitted videos by an accomplished mathematician, a link to a paper by another, and articulated several arguments of my own and you dismiss me as a dogmatist as if I'm simply being too hardheaded to debate???

Serendipper wrote:Below, you're proceeding to talk about what you haven't defined. What do you mean when you say there are "infinite" numbers between 1 and 0 if infinity is not boundless?

As I say, if you deny that there are infinitely many real numbers between 0 and 1, that is your choice. I assume you must deny there are infinitely many rational numbers between 0 and 1 as well. This is your right, to adopt a finitist or ultra-finitist stance. It just makes conversation pointless.

You still haven't defined what you mean by infinity. You say there are an unbounded number of numbers bounded by 1 and 0 and use that to prove that infinity has a bound

When you were in high school and they showed you the real number line in Algebra I when you were 14 years old or so, did you complain that there can only be finitely many points between 0 and 1? I confess I don't understand this point of view.

I understand the concept of dividing eternally, but that is never anything that can be actualized because when you stop to inspect how you're progressing, you will always have a finite amount of divisions between 1 and 0. When the day comes that forever has arrived, then you will have infinite numbers of divisions.

But really, if you want to use your time on God's earth to post to an Internet forum that you think there are only finitely many points on the real number line between 0 and 1, more power to you. I won't stand in your way.

This is your way of conceding?

Serendipper wrote: The first step in an argument is to define terms clearly before using the terms to construct an argument, otherwise you're saying a+b+c=z but not saying what the variables are. Defining terms commits you to what you're saying.

Mathematically, a set is infinite if it may be bijected to a proper subset of itself. That was one of your dictionary definitions if I recall. Galileo noted this in the 1600's and various non-Western mathematicians noted it in the 1200's or earlier.

That's just a more complicated way of saying boundless because the only way bijection would work is if both sets were boundless. The only reason 1,2,3,4.... and 2,3,4,5.... can correspond is if neither have an end.

Serendipper wrote:Probably why 1+2+3+4+5+.... forever = -1/12

Oh my. You are a victim of some very unfortunate misinformation floating about the Internet.

The infinite series 1 + 2 + 3 + 4 + ... of course diverges.

ONLY if you stop before forever has arrived. 1-1+1-1+1-1+1.... will always be either 1 or 0 unless you go to infinity in which case it converges to 1/2. Likewise 1+2+3+4+5... diverges unless you go to infinity in which case it converges to -1/12. It's been proven using your bijection rule.

Or we can say that it converges to +infinity in the extended real numbers.

No, not at infinity it does not. It doesn't even converge to infinity at a finite location.

Or we can say that it converges to Aleph-null as a cardinal, or omega as an ordinal.

Fantasy land.

These are all different ways of saying that this infinite sum "converges to a point at infinity."

Converging to infinity doesn't make sense and is like saying convergence to divergence. Points are unbounded (dimensionless) and so is infinity which argues for my position that neither zero nor infinity exists.

So, what is this -1/12 business about? Briefly, everyone knows that the sum of the positive integers is infinity, or undefined, whichever formalism you're using at the moment.

No, everyone doesn't know that. Everyone knows the sum of numbers produces a sum that is defined and finite. How can we have a sum that is undefined?

But there's a thing called "Zeta function regularization" that says that something called zeta(-1) is -1/12. And you can choose to INFORMALLY interpret zeta(-1) as the series 1 + 2 + 3 + ... But it is NOT the same thing as that series. I'll give you some links that explain all this.

We do not need the zeta function to prove 1+2+3... = -1/12. All we need is your definition of infinity (bijection).

Why some ignorant Youtubers decided to start an online campaign to confuse people I have no idea. But that's modern life. The Internet, which was supposed to make us all smart, has in fact made us all much more stupid. And this -1/12 business is a classic example.

You're the one who is in error, so you'd be privy to explanation as to why.

Serendipper wrote:infinity doesn't exist. If it did, the sum of all natural numbers would certainly equal it.

Oh my. As I said, it's perfectly ok for you to reject completed infinity That's finitism. You can even reject potential infinity. That's ultra-finitism.

I don't care about labels. I would call myself a realist if that didn't inject notions that things can be real without an observer.

Those positions are logically consistent (though there's no known axiomatic basis for ultra-finitism). But they're nihilistic, in the sense that you have just kicked the rug out from not only math, but physics as well. I'd say that's an intellectual challenge for your position, to rebuild physics on finitistic or even constructive principles. There actually are people doing the latter.

You're accusing me of what you're doing, My position is grounded in reality (physics, what really exists) while yours is off in some mathematical construct of imagination (hallucination) with no empirical footing whatsoever.

Every mathematician knows that 1 + 2 + 3 + ... diverges to infinity.

Every mathematician knows it converges to -1/12 at infinity or they wouldn't be a mathematician.

The interpretation of that sum as -1/12 is a distortion and abuse of a very sophisticated bit of mathematics in higher complex number theory. Somebody made a misleading video and a horde of people just ran with the wrong idea. It's awful frankly. But if that's your evidence for rejecting infinity, your argument is refuted. Please read the links I gave you above to put the -1/12 business into its proper context.

Did you read the links I gave? If you had, you would have realized the zeta function is not required to prove -1/12. The zeta function can be considered offtopic and inconsequential.

More nonsense on the same lines I imagine. The explanation of -1/12 in terms of analytic continuation and zeta function regularization is all over the Internet. For you to choose the completely wrong interpretation and then expound on your thesis of infinity based on it is truly ... well, it's your right. But I question your motivation. After all, the correct information is just as easy to find as the falsehoods.

The proof is demonstrated right before your eyes, yet you dismiss it because it transgresses your dogma

Oh that's right... you didn't look. Well let me force you to look:

S1 = 1-1+1-1+1-1....S2 = 1-2+3-4+5-6....S = 1+2+3+4+5...

S1 = 1/2

2S2 = bijection of itself, so...

1-2+3-4+5-6...0+1-2+3-4+5....

Which = 1-1+1-1+1-1.... which = 1/2

So S2 = 1/4

Now S - S2 =

1+2+3+4+5...-1+2-3+4-5...

Which = 0+4+0+8+0....

Which = 4(1+2+3+4+5....) = 4S

So S - S2 = 4S

S2 = 1/4

So S - 1/4 = 4S

S = -1/12

Now what will you say?

Serendipper wrote:You're making divisions where there are none, or did you forget that you cut it? You're creating differentiations on the fly and will need an infinite amount of time to finish, so you're assuming infinity in your proof of it. This was spelled out nicely in the article I posted http://theorangeduck.com/page/infinity-doesnt-exist

Here you're again arguing that the unit interval's not an infinite set. That's not a serious intellectual position.

No I'm arguing the delineations only exist in your imagination and are not existent in reality and in order to prove what exists in your mind would require infinite time.

Serendipper wrote:Keep Adding OneThe obvious argument for infinity existing is that, given some number, it is always possible to add one to get a new number. Therefore there must be an infinite number of numbers.

The problem with this argument is that it presupposes infinity exists already. More specifically it assumes that a process can be repeated an infinite number of times. If you can't repeat a process infinitely, and there isn't infinite time, it isn't possible to continue adding one forever.

If you reject taking successors, that makes you an ultra-finitist. Fine with me. You not only reject the real numbers but the counting numbers as well. Whatever. So what happens to physics in your theory? You can't do modern physics without the real numbers and the modern theory of infinite series.

Physics is finite and I've no clue to what physics you're referring. In the physics textbook it says 1+2+3+4+5... = -1/12

Serendipper wrote:Likewise, you're just adding new numbers between 0 and 1 and you're reliant upon infinite time to complete the process, but if you stop to inspect along the way, the answer will always be finite.

The real numbers are a mathematical abstraction. They don't require time or space or energy or any physical resource to construct.

So that means they don't exist.

I would agree with you that PHYSICAL infinity may well not exist. I don't believe it does. But we are talking about MATHEMATICAL infinity

.That's good enough for me.

Serendipper wrote:That is why 1-1+1-1+1-1... forever = 1/2 because if you stop, you get either 0 or 1.

It's tragic that people can get so many bad ideas from the Internet, yet won't take the time to learn any actual math.

Why are you patting yourself on the back when you're the one who is in error? 1-1+1-1+1-1... = 1/2 at infinity

Serendipper wrote:No, you've defined it to be so. You're not observing, but conjuring.

Did you have a bad math teacher when you were young? Where is this coming from?

Where are these ad homs coming from?

Serendipper wrote:You've defined the universe to be a fraction 1/x where x >1, so you've just moved the goal posts from simply counting forever to counting forever and plugging into a function... it's the same problem of using infinity to prove infinity. You need an infinite amount of x to prove there are infinite amounts between 1 and 0. Why bother? Just prove there are infinite x >1.

That, I'm afraid, is incoherent.

That reveals more about you than me. You need a set of infinite numbers in order to prove there are infinite numbers between 1 and 0, so instead of all that trouble, why not just prove there are infinite numbers?

I say the biggest number is the number of planck cubes in the observable universe and you cannot prove me wrong because there is no more room in the universe to do so. You could say if the universe were bigger, then a larger number could exist. Yeah but it's not, so it doesn't, and if it were, then it would still be finite. No computer is large enough; no imagination is large enough; no universe is large enough to hold an infinite number.

Serendipper wrote:You can only cut something so many times before the knife becomes larger than the thing you're cutting and that fact will become self-evident once you begin and persist in cutting.

The only way to "have" infinity is to stop the process, but if you stop the process, the result will always be finite.

How did you get these ideas? I'm curious how someone goes down this path of rejecting all of modern math and physical science.

I'm not rejecting physical science, but appealing to it instead of the fantasy of math.

For example, a road could be endless in one direction, but endful in another.

I attacked this in the OP.

Infinities cannot have beginnings or ends because those are boundaries and we said in the beginning that infinity has no boundaries. We cannot divide infinity in half and say infinity is bounded by this finite location and extends to infinity in that direction; it's nonsense and breaks our definition of infinity being boundless. Zero is not a boundary, but is just an arbitrary starting point on an infinite number line extending in both directions and we could just as easily started at -2,-1,0,1,2,3,etc or 5,6,7,8,etc. A line that is not infinite is a segment because all lines are defined to be infinite within the construct of mathematics; therefore a line with a beginning (such as a timeline) is not an example of infinity.

You: Here is a road that starts here and goes to infinity.Me: We could add more road, so it's not infinite.You: What do you mean?Me: Where you're standing... we could add more road and since we could add more, then it's not infinite, but bounded by the location where you're standing. If we could add more road, then the amount of road we have is not infinite.

MEST could in fact be infinitely divisible, but finitely multipliable, if you know what I mean, or vice versa.

I don't know what MEST is.

Just because there's an infinite amount of total apple, doesn't mean apple has to be omnipresent.

Yes it does because if there is a place that we could add another apple, then the amount of apples is not infinite. If there are infinite apples, then apples exist in every place there is to exist.

Now whether anything/everything actually is infinite, is another matter.I think finitude and infinitude are equally possible.But finite, unomniscient beings such as ourselves couldn't definitely prove either-or/neither-nor.But if something appears to go on, and on...and on, we may feel like concluding it's endless.Conversely, after many unsuccessful attempts to divide a particle, we may may like concluding it's indivisible.

We can divide down to quarks, but to divide a quark pair will create another quark pair from the energy used to divide them. There is no such thing as smaller than a quark because smaller than a quark is smaller than the thing that defines size.

Serendipper wrote:Below, you're proceeding to talk about what you haven't defined.

ps to my previous post. I did a quick preliminary read of your paper that you linked. I am impressed. You have a thesis. You are wrong about a lot of things, but you did mention all the relevant topics. You've done your homework.

Thanks! I've been debating infinity for years and finally decided to consolidate information into one place to avoid rehashing the same old arguments eternally.

From what I can see so far, you keep saying you're talking about mathematical infinity, but as arguments against it you immediately cite physical analogies. But just because something is physically impossible doesn't mean we can't work with it as an abstract entity. You constantly fall back on physical arguments even when you claim to be arguing against abstract mathematical infinity.

Well, I've defined infinity as boundless and asserted that what is without boundaries is not a thing neither in reality nor imagination. Just like i (sqrt -1), infinity may have productive uses, but likely it's indicative of some underlying mechanism that we don't yet understand.

You are absolutely correct that the nub of the matter is the Axiom of Infinity. It is indeed arbitrary, in the sense that both it and its negation are perfectly consistent with the other axioms of math. There is no absolute truth of the matter; and no logical reason to prefer one to the other.

However, there is a pragmatic, practical reason. When you assume the axiom of infinity, you can construct the real numbers and do all of modern math and physics. When you deny the axiom of infinity, you get a far more paltry universe that can't be made to serve the needs of mathematical foundations.

I'm not sure this is true because we routinely use PI, but never all infinite digits of it.

The axiom of infinity is no more "true" or "false" than whether the knight in chess "really moves that way." What a ridiculous question! It's a formal game. It's not true and it's not false. The rules are what they are, and if they are consistent and interesting, we accept them. Chess is fun to play, so we play. Math is fun to play, and the physicists and engineers find it useful. That's as far as the ontology goes.

Right, the concept of "knight" is only relative to the game of chess and the concept of truth is only relative to the duality of the universe.

I mean, forget about an infinite set. Does the empty set exist? Ponder that.

No, the empty set does not exist unless it contains potential, but then it wouldn't be empty. I'm saying neither zero nor infinity exists and they are equally absurd.

I hope you'll take some of my points to heart. You're tilting at windmills. You think somebody thinks the axiom of infinity is true. On the contrary. People who think about the question at all, understand that the reason we accept infinity in math is because it's useful.

How is infinity useful? I googled it and here is what I came up with:

We Don’t Need the InfiniteLet’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 10^80 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.http://blogs.discovermagazine.com/crux/ ... 8soKiXwaHs

ps -- You may be interested to read a pair of papers by Penelope Maddy, Believing the Axioms parts I and II. You can find pdfs if you Google around. She walks through each axiom of ZFC and discusses the history and philosophy of how and why it came to be adopted.

barbarianhorde wrote:How... is it possible to argue against mathematical infinity?

To assume infinities is to concede discontinuities exist in nature as if an asymptotic curve could exist in nature and that y=1/x wouldn't connect at x=0. The truth is far more likely that the Cartesian coordinates are better represented on a sphere rather than infinite plane. Math reflects reality to some degree less than 100%.

Derek Abbott, Professor of Electrical and Electronics Engineering at The University of Adelaide in Australia, has written a perspective piece to be published in the Proceedings of the IEEE in which he argues that mathematical Platonism is an inaccurate view of reality. Instead, he argues for the opposing viewpoint, the non-Platonist notion that mathematics is a product of the human imagination that we tailor to describe reality.

This argument is not new. In fact, Abbott estimates (through his own experiences, in an admittedly non-scientific survey) that while 80% of mathematicians lean toward a Platonist view, engineers by and large are non-Platonist. Physicists tend to be "closeted non-Platonists," he says, meaning they often appear Platonist in public. But when pressed in private, he says he can "often extract a non-Platonist confession."

So if mathematicians, engineers, and physicists can all manage to perform their work despite differences in opinion on this philosophical subject, why does the true nature of mathematics in its relation to the physical world really matter?

The reason, Abbott says, is that because when you recognize that math is just a mental construct—just an approximation of reality that has its frailties and limitations and that will break down at some point because perfect mathematical forms do not exist in the physical universe—then you can see how ineffective math is.

And that is Abbott's main point (and most controversial one): that mathematics is not exceptionally good at describing reality, and definitely not the "miracle" that some scientists have marveled at. Einstein, a mathematical non-Platonist, was one scientist who marveled at the power of mathematics. He asked, "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"

In 1959, the physicist and mathematician Eugene Wigner described this problem as "the unreasonable effectiveness of mathematics." In response, Abbott's paper is called "The Reasonable Ineffectiveness of Mathematics." Both viewpoints are based on the non-Platonist idea that math is a human invention. But whereas Wigner and Einstein might be considered mathematical optimists who noticed all the ways that mathematics closely describes reality, Abbott pessimistically points out that these mathematical models almost always fall short.

"I argue that there are many more cases where math is ineffective (non-compact) than when it is effective (compact). Math only has the illusion of being effective when we focus on the successful examples. But our successful examples perhaps only apply to a tiny portion of all the possible questions we could ask about the universe."

Is there a way of proving there can be only a finite number of numbers?

What do you mean by "be"? How can something be if it has no boundaries?

What do you mean by "prove"? Do you mean arbitrarily define some axioms and then show how a conclusion fits?

It is basically the time taken by light of specified wavelength emitted by a Cesium 133 atom to execute 9,192,631,770 vibrations.

Right after the big bang there was no cesium 133, so how long was 1 second?

Cesium 133 is part of one of the ways we measure time. Of course we experience time without cesium 133 and we don't know if there were thing to relate one thing to another back then. We do not know if universes emerge from a larger universe - one of the theories out there - and so this larger context would have time.

We cannot experience time if there is nothing defining time. We cannot experience time if there is nothing to relate to. Time has absolutely no definition absence of relation. All time means is a relation of one thing to another thing. Time is not fundamental and there is no such thing as objective time. We don't even need to go before the big bang to see this is true, but merely AT the big bang will suffice since at that theoretical point of "pointness" there was no delineation of things in order to have one thing relating to another thing, but only unity. In fact, it's still that way since every emission and reception of a photon is a single timeless event meaning that every point in spacetime is still joined as one ( as evidenced by the lack of time and distance between them).

Time has no intrinsic meaning but is only a relation of one thing to another thing. If it helps to see a phd physicist and philosopher say the same thing, fwd to 38:00 here https://www.youtube.com/watch?v=N-NTXoYTvao

(Btw the consensus of those 4 guys is that time is not fundamental. The guy on the right however disagrees somewhat because if the laws of the universe are emergent, then they must have evolved through time and therefore time could not be a product of the fundamental laws, yet he still doesn't see time as truly fundamental.)

That's all peachy, but again, in the astrophysics community there is no consensus about infinite vs. finite, in time or volume.

Lack of consensus doesn't mean anything, but you seem to like consensus, so I figured you'd appreciate the fact that the 4 physicists agree that time is not fundamental. I'm not appealing to the consensus and it wouldn't bother me if all 4 were in unanimous disagreement with me; it's beside the point and why I said Btw (by the way).

There could be a seemingly infinite number of seconds in the finite span between now and the big bang just like there are seemingly infinite numbers of numbers between 0 and 1 because time is related to the very thing that was created and what we call seconds are changing the closer we get to the beginning (light moves faster), but this infinity is only an illusion produced by infinite regression once again because seconds are defined by the thing we're trying to measure.

Could be. I am not arguing that the universe is infinte, though I tend to think it is, but that's just my intuition. But, sure could be. I am not saying you are wrong. I am questioning your certainty or better put, I see little reason to be certain around this issue.

The infinite is not anything that can exist, so the fact that the universe appears infinite means infinite regression of self-inspection (ie the thing doing the looking is the thing being looked at). There is no other explanation. It's not an argument from ignorance as if because I can't think of another explanation, then this one will do. No, it's that there cannot be another explanation because the actuality of infinity cannot exist. If the universe is infinite, then it is an illusion produced by the universe beholding itself.

Asking what came before the big bang is akin to asking what's north of the north pole: there is no north of the north pole and there is no before the beginning.

Seriously, I knew this idea 20 years ago and I am so tired of it getting trotted out by other laypeople. Yes, it seemed like there was some consensus for a while about this, but there isn't now. So trotting out old explanations that scientists and science writers used to try to help people with this counterintuitive idea THEN, does very little for me, because they are not sure about this at all.

You're bragging that? All it means is you didn't understand it then and 20 years has not helped. Once you get your head around the fact that time is relational and only relational, then everything should fall into place.

Truth is independent of majority consensus and, apart from grass being green and the sky blue, majority opinion is more likely to be wrong than right.

Right, but you are working with opinions from the scientific community and THEIR consensus.

No I'm not. I'm working with my own ideas. Sure it may have been their idea first, but I stole it and now it's mine. I don't hold their ideas because they said it, but because the ideas made sense to me.

Am I supposed to take a video with four people in it as evidence when you are here dismissing the consensus of the scientific community? A consensus of four is also fallible.

I said "If it helps to see a phd in physics and philosophy say the same thing, then watch the video." If it wouldn't help, then don't. I'm not appealing to them, but trying to help you. I tell ya what... don't watch and instead pretend you already know

And this holds for what is below. Your ideas are based on and specifically refer to ideas trickling out of the scientific community. Now you are telling me that we shouldn't look at consensus in the science community.

No, we take what works and not because the consensus agrees, but because the idea is sensible. The amount of people in agreement means nothing.

Since I make no claims to be sure, that's really easy to live with. Since you seem to think you have proven the universe is finite, good luck with having undermined your own authorities that you appeal to.

The only authority I appeal to is reason and I'm not sure what you're on about.

Atheism is the prevailing bias and physicists will not let go of infinity until theism is eliminated because they need it as a substitute for god.

But many physicist let go of infinity. Now you're just making stuff up.

But you just said there is no consensus. If there is no consensus, how do you know many physicists let go of infinity? Doesn't "many" = consensus?

Does consensus mean more than half? Have more than half the physicists let go of infinity? In that case, there is consensus that the universe is finite.

Does consensus mean 100%? In that case, they're still clinging to infinity because they aren't unanimous.

The fact remains that the scientific community (not limited to just physicists) does not like the idea of god, so infinity is an easy way around it and I couldn't agree that scientists are approaching the problem objectively so long as theists exist in any meaningful numbers, therefore they will always be biased with an affinity towards infinity until there is reason not to be.

So (how did you put it...) cornering an astrophysicist at a party for his/her opinion about consensus opinion + $2 will get you a cup of brown liquid.

Right except I also researched what percentages believed various positions and this fit with his estimates of his peers. He wasn't telling me his stand, I don't know what that is in fact.

Percentages are irrelevant.

And again, you are the one claiming to have demonstrated that infinity is impossible. I am saying that I am skeptical.

You're skeptical that something without boundaries cannot be considered a thing? Then what could define a thing absent boundaries? Why would you be skeptical without already harboring some alternative view? You must have some conceptualization of a borderless thing in order to assert its plausible existence. IOW, you wouldn't assert a squared circle without having some idea of how it might exist.

Fine, now you seem to be saying you do not base your ideas on the ideas of astrophysicists and cosmologists and don't care what the experts say. Great. I see even less reason to believe your ideas which are in part based on the positions of physicists and cosmologist. IOW their ideas are parts of your arguments.

I've made them my own... and not because they said it, but because the ideas are sensible. If you don't want to believe me, then we'll put you down as disagreement for lack of authority: I don't have authority to speak, so whatever I said can be disregarded.

You seem to think you can demonstrate infinity is not possible. I just don't see it.

So you admit your lack of perception? I agree, so we have consensus that you can't see

Especially now that you dismiss scientists, which would include those whose ideas you use.

I dismiss them as authorities yet accept the ideas that seem sensible to me.

I think we should remain utterly unconvinced by your arguments, only more so now.

Yes, well, be that as it may, I'm not convinced that the audience would come to the same conclusion. If I am reading a series of comments and one guy says the other hasn't done a good job, it doesn't sway me a bit; only information sways me.

this is going nowhere, we need less rhetoric and more technical analysis people.

Can you start over, but with in mind what these guys have to say? Then it will be more substantial and involuntary deceit can't be as big a part. I mean misreadings and stuff. A n n o y I n g .viewtopic.php?f=4&t=185099&p=2470116

Oh yeah and lets follow definitional logic, so first define what there are supposed to be either infinitely or finitely many of.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

So if you go on to define infinity as a thing and calculate with it, thats faux because you're working with a limited thing that is different from something else, so its not actually infinite. Its just a vector with the presumption that it won't run into a limit. But I think Serendipper is prob right that this sort of infinity is actually a circle. Even because as the range progresses away from the definer its steps become less significant and thats already a curving.

It is true that liberty is precious; so precious that it must be carefully rationed. ~ Владимир Ильич Ульянов Ленин

According to Spinozas logic here, a substance can only be conceived as infinite, the argument materializes in proving proposition VIII.

[list]"DEFINITIONS.

I. By that which is self—caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent.

II. A thing is called finite after its kind, when it can be limited by another thing of the same nature; for instance, a body is called finite because we always conceive another greater body. So, also, a thought is limited by another thought, but a body is not limited by thought, nor a thought by body.

III. By substance, I mean that which is in itself, and is conceived through itself: in other words, that of which a conception can be formed independently of any other conception.

IV. By attribute, I mean that which the intellect perceives as constituting the essence of substance.

V. By mode, I mean the modifications[1] of substance, or that which exists in, and is conceived through, something other than itself.

[1] "Affectiones"

VI. By God, I mean a being absolutely infinite—that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.

Explanation—I say absolutely infinite, not infinite after its kind: for, of a thing infinite only after its kind, infinite attributes may be denied; but that which is absolutely infinite, contains in its essence whatever expresses reality, and involves no negation.

VII. That thing is called free, which exists solely by the necessity of its own nature, and of which the action is determined by itself alone. On the other hand, that thing is necessary, or rather constrained, which is determined by something external to itself to a fixed and definite method of existence or action.

VIII. By eternity, I mean existence itself, in so far as it is conceived necessarily to follow solely from the definition of that which is eternal.

Explanation—Existence of this kind is conceived as an eternal truth, like the essence of a thing, and, therefore, cannot be explained by means of continuance or time, though continuance may be conceived without a beginning or end.

AXIOMS.

I. Everything which exists, exists either in itself or in something else.

II. That which cannot be conceived through anything else must be conceived through itself.

III. From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow.

IV. The knowledge of an effect depends on and involves the knowledge of a cause.

V. Things which have nothing in common cannot be understood, the one by means of the other; the conception of one does not involve the conception of the other.

VI. A true idea must correspond with its ideate or object.

VII. If a thing can be conceived as non—existing, its essence does not involve existence.

PROPOSITIONS.

PROP. I. Substance is by nature prior to its modifications.

Proof.—This is clear from Deff. iii. and v.

PROP. II. Two substances, whose attributes are different, have nothing in common.

Proof.—Also evident from Def. iii. For each must exist in itself, and be conceived through itself; in other words, the conception of one does not imply the conception of the other.

PROP. III. Things which have nothing in common cannot be one the cause of the other.

Proof.—If they have nothing in common, it follows that one cannot be apprehended by means of the other (Ax. v.), and, therefore, one cannot be the cause of the other (Ax. iv.). Q.E.D.

PROP. IV. Two or more distinct things are distinguished one from the other, either by the difference of the attributes of the substances, or by the difference of their modifications.

Proof.—Everything which exists, exists either in itself or in something else (Ax. i.),—that is (by Deff. iii. and v.), nothing is granted in addition to the understanding, except substance and its modifications. Nothing is, therefore, given besides the understanding, by which several things may be distinguished one from the other, except the substances, or, in other words (see Ax. iv.), their attributes and modifications. Q.E.D.

PROP. V. There cannot exist in the universe two or more substances having the same nature or attribute.

Proof.—If several distinct substances be granted, they must be distinguished one from the other, either by the difference of their attributes, or by the difference of their modifications (Prop. iv.). If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute. If by the difference of their modifications—as substance is naturally prior to its modifications (Prop. i.),—it follows that setting the modifications aside, and considering substance in itself, that is truly, (Deff. iii. and vi.), there cannot be conceived one substance different from another,—that is (by Prop. iv.), there cannot be granted several substances, but one substance only. Q.E.D.

PROP. VI. One substance cannot be produced by another substance.

Proof.—It is impossible that there should be in the universe two substances with an identical attribute, i.e. which have anything common to them both (Prop. ii.), and, therefore (Prop. iii.), one cannot be the cause of the other, neither can one be produced by the other. Q.E.D.

Corollary.—Hence it follows that a substance cannot be produced by anything external to itself. For in the universe nothing is granted, save substances and their modifications (as appears from Ax. i. and Deff. iii. and v.). Now (by the last Prop.) substance cannot be produced by another substance, therefore it cannot be produced by anything external to itself. Q.E.D. This is shown still more readily by the absurdity of the contradictory. For, if substance be produced by an external cause, the knowledge of it would depend on the knowledge of its cause (Ax. iv.), and (by Def. iii.) it would itself not be substance.

PROP. VII. Existence belongs to the nature of substances.

Proof.—Substance cannot be produced by anything external (Corollary, Prop vi.), it must, therefore, be its own cause—that is, its essence necessarily involves existence, or existence belongs to its nature.

PROP. VIII. Every substance is necessarily infinite.

Proof.—There can only be one substance with an identical attribute, and existence follows from its nature (Prop. vii.); its nature, therefore, involves existence, either as finite or infinite. It does not exist as finite, for (by Def. ii.) it would then be limited by something else of the same kind, which would also necessarily exist (Prop. vii.); and there would be two substances with an identical attribute, which is absurd (Prop. v.). It therefore exists as infinite. Q.E.D. " --- Ethica

barbarianhorde wrote:this is going nowhere, we need less rhetoric and more technical analysis people.

Can you start over, but with in mind what these guys have to say? Then it will be more substantial and involuntary deceit can't be as big a part. I mean misreadings and stuff. A n n o y I n g .viewtopic.php?f=4&t=185099&p=2470116

Oh yeah and lets follow definitional logic, so first define what there are supposed to be either infinitely or finitely many of.

I don't know what can be gained with that thread because the only one who really asserted anything was James and he was wrong. We know from experimentation that we can't split quark pairs without creating new quarks from the energy put into splitting them, so there is no way to cut down to a smaller size and certainly no way to see it since we can't use electrons to look at things smaller than electrons.

barbarianhorde wrote:So if you go on to define infinity as a thing and calculate with it, thats faux because you're working with a limited thing that is different from something else, so its not actually infinite. Its just a vector with the presumption that it won't run into a limit. But I think Serendipper is prob right that this sort of infinity is actually a circle. Even because as the range progresses away from the definer its steps become less significant and thats already a curving.

Loss of significance seems like a good theory to me.

barbarianhorde wrote:Also we need to compare infinity to something, its in-finite, it stands opposed to the finite range of some form or something.It is not a comprehensible idea on its own.

Infinite to what?

(I know that sounds strange but you gotta bend your thought, because thats in the end what its all about. Can the thought of infinity exist together with the though of existence )

If we stick to the definition of the infinite as unbounded, then the opposite is the bounded.

Fixed Cross wrote:According to Spinozas logic here, a substance can only be conceived as infinite, the argument materializes in proving proposition VIII.

[list]"DEFINITIONS.

I. By that which is self—caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent.

What's the definition of existence?

Existence is an awareness of a being of everything apparent and conceivable.

Since it is realized through living, that existence is only a partial appearance of that everything conceivable, and even that is only a part of a totality of all, that becomes manifest, that totality is an unreachable absolute.

It is through the door of perception , that absolute manifests from birth to death.

It is that absolute that we as humans have to traverse into incarnation and re-incarnation. Every existence is another manifestation of that Absolute.

Serendipper wrote:If we stick to the definition of the infinite as unbounded, then the opposite is the bounded.

I've already given you examples of bounded infinite sets. The closed unit interval [0,1] is infinite and bounded. It also happens to contain its upper and lower limit points, 0 and 1. Even its cardinality is bounded (a point you are confused about in your earlier reply to me), since its cardinality is that of the reals, which is strictly smaller than the cardinality of the powerset of the reals by Cantor's theorem.

Why do you insist on using a definition that's demonstrably wrong? You can't fall back on,"Oh it's in the dictionary," since the dictionary is not authoritative on technical matters. You enjoy making up your own definitions, but that convinces no one other than you.